If all illegal moves were permissible, then people would lose time all of the time on mouseslips and fat fingering
Why Can't you play illegal moves?

If all illegal moves were permissible, then people would lose time all of the time on mouseslips and fat fingering
If all illegal moves were permissible, then people would lose time all of the time on mouseslips and fat fingering
That's their own fault for not being coordinated enough to move a chess piece using a mouse or touch screen.

If you were to realize this idea, I’d think you would have to make two categories of games: ones with illegal moves and one without

This is not a good idea. This idea would make all games a draw because for example, in the above position, black is not checkmated because he is allowed to play an illegal move.

I think cheating in general should be allowed, you should be able to cheat because it's your opponents fault for not noticing, not yours for doing it. I mean come on, the whole point of games is to try to win by cheating, why would you actually try to get good at it, or care if other people have fun playing you?

Illegal moves in Scrabble are programmed that way for a good reason. In chess, a player always knows which moves are legal or not, but this isn't always the case in Scrabble. An inferior Scrabble player will not always be aware of potential legal moves they have. This player would benefit from a game that automatically block illegal moves, because that player could "guess" new words by playing words they hope that are legal moves. With the challenge system, players that know fewer words are properly penalized for their lack of skill, and the balance of power tilts towards those who have a bigger vocab.
There is no reason to implement the same "illegal move" approach in chess. In theory, both players always know when a move is legal or not; the only time illegal moves are played is when a player does not notice it. Allowing such moves doesn't add to the game of chess at all.
In real life, illegal moves are part of the game. If I make an illegal move, you can challenge me and if you are correct, you gain 2 minutes on your clock. This is similar to scrabble, some sites allow you to play fake words, hoping to bluff your opponent. It can backfire if your opponent challenges. Why can't we implement this in online chess?